Vectors & Space Geometry (Part 2)

Question 1

What’s a cylinder in the space (general)?

Answer 1

This depends on a curve on a plane and a line not parallel to the curve.

It is the geometric figure generated by the line that moves parallel to L and passes through the curve C

Question 2

What is a trace and provide an example?

Answer 2

A trace is a the intersection between a curve with one of the coordinate planes

z = x^2 —> Here, y = 0. Thus, the trace of this curve is a parabola generated along the y axis

Question 3

Mention most known quadric surfaces:

Answer 3

Ellipsoid
Hyperboloid of 1-sheet
Hyperboloid of 2-sheet
Elliptic Cone
Elliptic Paraboloid
Hyperbolic Paraboloid

Question 4

What are the equations involved when revolving a curve around a given axis?

Answer 4

1) x^2 + y^2 = (f(z))^2 around z
2) x^2 + z^2 = (f(y))^2 around y
3) z^2 + y^2 = (f(x))^2 around x

1) x or y equal to f(z)
2) x or z equal to f(y)
3) z or y equal to f(x)

Question 5

How do you pass from Cartesian (x,y,z) to Cylindrical Coordinates?

Answer 5

By a new format (r, theta, z)

r can be found using Pythagoras and theta using arctangent of y divided by x

Question 6

How do you pass from Cartesian (x,y,z) to Spherical Coordinates?

Answer 6

Using the new format provided by (rho, theta, phi)

rho can be found using Pythagoras

theta = arctangent (y/x)

z = rho times cosine of phi

Question 7

Tell me some relationships related to r, rho, theta, phi:

Answer 7

z = rho * cos(phi)

r = rho * sin(phi)

x = rho*sin(phi)cos(theta)

y = rho*sin(phi)sin(theta)